Curvature bound for \(L_p\) Minkowski problem (Q6634752)

The authors consider the \(L_{p}\) Minkowski problem for a compact and convex set \(\Omega \) in \(\mathbb{R}^{n+1}\) with a nonempty interior, whose Gauss curvature is prescribed: \[S_{p}(\Omega ,\cdot )=\mu \] for a given measure \( \mu \) on \(\mathbb{S}^{n}\), where \(S_{p}(\Omega ,\cdot )\) is the \(L_{p}\) surface area measure of \(\Omega \), determined by the Aleksandrov variational formula \[\frac{dV(\Omega +_{p}t\cdot _{p}\Omega _{0})}{dt}\mid _{t=0^{+}}= \frac{1}{p}\int_{\mathbb{S}^{n}}u_{\Omega _{0}}^{p}(z)dS_{p}(\Omega ,z)\] for any convex body \(\Omega _{0}\), \(V(\Omega )\) being the volume of \(\Omega \), \( u_{\Omega _{0}}:\mathbb{S}^{n}\rightarrow \mathbb{R}\) the support function of \(\Omega _{0}\) defined by \(u_{\Omega _{0}}(z)=\max\{z\cdot x:x\in \Omega _{0}\}\), and \(t_{1}\cdot _{p}\Omega _{1}+_{p}t_{2}\cdot _{p}\Omega _{2}\) the \(p\)-linear combination of convex bodies defined as the convex body with support function \((t_{1}u_{\Omega _{1}}^{p}+t_{2}u_{\Omega _{2}}^{p})^{1/p}\). \N\NIn the paper under review, the measure \(\mu \) is supposed to have a density \(f\), that is \(d\mu =fd\sigma \), where \(\sigma \) is the spherical Lebesgue measure. The preceding \(L_{p}\) Minkowski problem can be interpreted as a Monge-Ampère type equation \[\det(u_{ij}+u\delta _{ij})=fu^{p-1}\] on \( \mathbb{S}^{n}\), where \(u\) is the support function of \(\Omega \), provided that \(u\) is positive. The support function \(u\in C(\mathbb{S}^{n})\) of a convex body \(\Omega \) or its convex hypersurface \(\Sigma =\partial \Omega \) is called a generalized solution to the Monge-Ampère equation if \(\Omega \) solves the \(L_{p}\) Minkowski problem \[\int_{E}dS_{p}(\Omega ,z)=\int_{E}fd\sigma \] for all Borel sets \(E\subset \mathbb{S}^{n}\). \N\NThe main result of the paper proves that if \(p\in (-\infty ,-n+2]\) and \(f\) is a positive and smooth function, then any generalized solution \(\Sigma \) to the \( L_{p}\) Minkowski problem is a hypersurface of class \(C^{1,1}\). Moreover, its principal curvatures are uniformly bounded by a constant depending only on \( n,p\), the diameter of \(\Sigma \), \(\left\Vert f\right\Vert _{C^{2}(\mathbb{S} ^{n})}\), and \(\min_{\mathbb{S}^{n}}f\). The authors prove that this result is sharp in the sense that if \(p\in \lbrack -n+2,1)\), then there exists a generalized solution \(\Sigma \) to the Monge-Ampère equation such that \( \Sigma \) is a hypersurface of at most class \(C^{1,\frac{1}{n+p-1}}\) and \(f\) is a positive smooth function. If \(p\in (1,n+1)\), then there exists a generalized solution \(\Sigma \) to the Monge-Ampère equation such that \(\Sigma \) is a hypersurface of at most class \(C^{1,\frac{n-p+1}{n+p-1}}\) and \(f\) is a positive smooth function. \N\NFor the proof, the authors use the notion of anisotropic \(\alpha \)-Gauss curvature flow, with \(\alpha =\frac{1}{1-p}\) and they analyze the local behaviour of such a flow when \(p\leq -n+2\). They also prove a uniform upper bound for the principal curvatures of \(\Sigma \).